The special values of the standard 𝐿-functions for 𝐺𝑆𝑝_{2𝑛}×𝐺𝐿₁

Abstract

We prove the expected algebraicity property for the critical values of character twists of the standard L L -function associated to vector-valued holomorphic Siegel cusp forms of archimedean type ( k 1 (k_1 , k 2 k_2 , …, k n ) k_n) , where k n ≥ n + 1 k_n \ge n+1 and all k i k_i are of the same parity. For the proof, we use an explicit integral representation to reduce to arithmetic properties of differential operators on vector-valued nearly holomorphic Siegel cusp forms. We establish these properties via a representation-theoretic approach.